Arithmetic-Geometric spectral radii of Unicyclic graphs

Abstract

Let $d_{v_{i}}$ be the degree of the vertex $v_{i}$ of $G$. The arithmetic-geometric matrix $A_{ag}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle \frac{d_{v_{i}}+d_{v_{j}}}{2\sqrt{d_{v_{i}}d_{v_{j}}}}$ if the vertices $v_{i}$ and $v_{j}$ are adjacent, and 0 otherwise. The arithmetic-geometric spectral radius of $G$, denoted by $\rho_{ag}(G)$, is the largest eigenvalue of the arithmetic-geometric matrix $A_{ag}(G)$. In this paper, the unicyclic graphs of order $n\geq5$ with the smallest and first four largest arithmetic-geometric spectral radii are determined.